The Seifert-van Kampen Theorem Example 2

On The Seifert-van Kampen Theorem page we stated the very important Seifert-van Kampen theorem. We will now look at some examples of applying the theorem. More examples can be found on the following pages:

• The Seifert-van Kampen Theorem Example 1

• The Seifert-van Kampen Theorem Example 2

• The Seifert-van Kampen Theorem Example 3

Example 2

Let $X$ be the torus with a line segment that touches the torus only at its endpoints as depicted below. Use the Seifert-van Kampen theorem to find the fundamental group of $X$.

The line across the hole of the torus can be deformed to:

Let $U_1$ and $U_2$ be the open sets depicted below:

We see that the circle is a deformation retract of $U_1$. We also see that the torus is a deformation retract of $U_2$. Lastly we see that a single point is a deformation retract of $U_1 \cap U_2$. Therefore:

(1)

\begin{align} \quad \pi_1(U_1, x) & \cong \mathbb{Z} = \langle \alpha : \emptyset \rangle \\ \quad \pi_1(U_2, x) & \cong \mathbb{Z} \times \mathbb{Z} = \langle \beta, \gamma : \emptyset \rangle \\ \quad \pi_1(U_1 \cap U_2, x) & \cong \{ 1 \} \end{align}

Therefore by the Seifert-van Kampen theorem we have that:

(2)

\begin{align} \quad \pi_1(X, x) \cong \langle \alpha, \beta, \gamma : \emptyset \rangle \end{align}